Sja Ch 8 meta notes

Sja Chapter 8 meta notes: volume forms PhL 8.23.15 Section 8.1. Volume of an n-piped in RN. [91] Theorem 8.3: The volume of an n-piped spanned by {ai} within N dimensional space is this (n ≤ N) volume of n-piped = where A = (a1, a2.....an) = N rows and n columns, so A = Nxn = tall I eventually will get this concept integrated into tensor doc. [ did that task on 5.8.16 in Section 8.4 (h)] I tried that today, but it is a larger subject than I at first thought, since I want to write something about faces of faces of faces in a vector sense. This could be a major overhaul of Appendix B. So don't do that right now! It will be an interesting tensor doc edit session when I get around to it. Definition 8.1: defining the volume of an n-piped in RN by a set of four axioms (typical Sjamaar): So here Sja writes four axioms and uses them to define volume in RN and that is what leads him to the above theorem. Corollary 8.4. In the case n = N, A is square so det(ATA) = [det(A)]2 and we get volume = det(A) which is my main tensor doc result for an N-piped in RN. Section 8.2. Orientations [94] In 2D this is the side of a square, in 3D this is the sign of det(A) where A = [e1, e2....en] . If you shuffle basis vectors, you can change the orientation as in [e2, e1....en] = - [e1, e2....en] . He first talks about a generic oriented vector space in this manner, but then you can treat TxM at each point x on a manifold as such an oriented vector space. For this tangent space, he likes the idea of adding the normal first in the list so you would have (n, v1....vn-1) where the last set of vectors spans the tangent space. In more detail: You know there is some well-defined set of +1 orientation [e1, e2....en] in Rn . Rotate things so that [e1, e2....en-1] spans TxM for some particular x. Note that en is already determined in direction. Now the normal at x is orthogonal to the other ei of TxM, we know that much. If we were to add the normal to the end of the list to get [e1, e2....en] = [e1, e2....en-1, n] , then you would select n = en for the direction of your normal! However, I guess convention says to put n at the start of the list. Then you must make the choice that n = (-1)n+1en, since in terms of orientation sign, [e1, e2....en] = [ (-1)n+1en, e1, e2....en-1] So this example is showing how you should choose the direction of the normal at point x! This was a question I raised above. I agree that you can map normals of a manifold to points on a sphere of the same dimension, which sphere points will have that normal. He calls this the Gauss Map of M. Idea of M being 2-sided or 1-sided but no mobius yet. Claim 8.8. If a manifold M is defined by a single smooth equation c = φ(x) where c is a regular point, then that manifold is orientable. Did not pursue this idea much. Mobius must violate this somehow. Section 8.3. Volume Forms [96] Here μ is the differential form for volume, but we don't have a general expression for it over all of M. In typical manifold fashion, we think of the manifold as a composite of mappings and then think about just one of those pieces or patches of M. Then for piece i we get [ μ = μM ] μi = ψi* μ = [ det{(Dψi)T(Dψi)}]1/2 dt1dt2....dtn // definitely a pullback statement! where ψi is the embedding function for patch i of the manifold. Notice the obvious ATA volume thing going on here. Identify A = (Dψ) whose column vectors are a basis of TxM. This is the "volume form" or perhaps the "measure" at point x on some manifold. It might be the area measure on a 2D surface. Sja shows in Theorem 8.10 that the above construct satisfies the consistency condition μj = (ψi-1o ψj)* μi ≡ φ*μi . where ψi-1o ψj ≡ φ Example 8.11. For the identity map ψ(t) = t for U→U We know that (Dψ) = 1 and then (Dψ)T(Dψ) = 1 and then the volume form page 96 E is just μi = 1 dt1....dtn, the obvious result. Example 8.12. Here f(t) is a scalar function and we take ψ(t) = (t,f(t)) which is that graph thing. We find that (Dψ)(t) = (1, f'(t)) (Dψ)T(Dψ) = (1, f'(t))T (1, f'(t)) = 1 + f'(t)2 μ = ψ*μM = [det((Dψ)T(Dψ))]1/2dt = dt So this is what you integrate to get the arc length of a curve. I show how this relates to wiki and Buck. Note that here t is really the x of (x,y) for a function y = f(x). This is different from Buck's s which is a parameter along the curve. Sja uses the word hypersurface to refer to a surface of dimension n-1 which lies in Rn. Wiki agrees with this usage, I guess I never really used that word strictly. For me, n-2 surface would also be a hypersurface if n-2 = 7, say. Comment 1. *dx as a product of differentials represents a volume element of M which has dim n-1 For example, if M has dim = 2 within R3, then *dx = an area (vector). The area elements would be (*dx)i such as (*dx)3 = dx1dx2 So (*dx) is an n-1 dimension volume element on M Theorem 8.14. In n dimensions we know what vector dx looks like, page 98 A. Each entry is a trivial 1-form. The Hodge * acts on these to give "all the others" and so the rest of A makes fine sense. The theorem makes this claim: F (*dx) = (Fn) μM For a regular 2D surface in E3 this is basically the F dA thing where we want to integrate over area of the surface. I usually write this as F dA = (F ) dA and so this theorem is just saying that dA = μM which is the "area form" in my example here. He gives two proofs of theorem 8.14, I follow through only on the first. Set F = n to get n *dx = μM like n dA = dA // this is page 99 A Example 8.16. Consider hypersurface φ(x) = c where c = scalar = regular value of φ. Then n = *dx = μM dx = μM For a sphere, φ(r) = |r| = r, φ = and the above says *dx = μM or dx = μM and n =